Theorem sbalv 2204

Description: Quantify with new variable inside substitution. (Contributed by NM, 18-Aug-1993.)

Hypothesis

Ref	Expression
sbalv.1	⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)

Assertion

Ref	Expression
sbalv	⊢ ([𝑦 / 𝑥]∀𝑧𝜑 ↔ ∀𝑧𝜓)

Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥, 𝑦, 𝑧)   𝜓(𝑥, 𝑦, 𝑧)

Proof of Theorem sbalv

Step	Hyp	Ref	Expression
1		sbal 2203	. 2 ⊢ ([𝑦 / 𝑥]∀𝑧𝜑 ↔ ∀𝑧[𝑦 / 𝑥]𝜑)
2		sbalv.1	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
3	2	albii 1848	. 2 ⊢ (∀𝑧[𝑦 / 𝑥]𝜑 ↔ ∀𝑧𝜓)
4	1, 3	bitri 278	1 ⊢ ([𝑦 / 𝑥]∀𝑧𝜑 ↔ ∀𝑧𝜓)

This proof depends on syntax axioms:   ↔ wb 209  ∀wal 1567  [wsb 2095

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-11 2191

This proof depends on definitions:  df-bi 210  df-an 401  df-ex 1809  df-sb 2096

This theorem is used by:  sbex  2315  sbmo  2641  sbabel  2956  mo5f  32846